# The SABR model: theory and application

Søren Skov Hansen

Student thesis: Master thesis

## Abstract

This thesis is based on the market observable volatility smiles for swaptions. We present different ways of handling these smiles and discuss the models as well as their implications. Initially, we introduce the financial theory that will be fundamental to our thesis. Following, we move on to a brief empirical study of volatility, where we show that volatility, contrary to the assumption made in the classical Black-Scholes setup, is constant neither in time nor in strike price. The first model we analyze is the the local volatility model. We discuss the model and give a numerical example on how the model can be calibrated to fit an observed volatility smile. Ultimately, we look into the inherent dynamics of the local volatility model, and we find that it has the counterintuitive property of shifting the volatility smile in the opposite direction of the price of the underlying asset when this shifts. Our second and main model is the SABR model. We present the model, and examine how its parameters influence the shape of the fitted volatility smile. Following, we investigate the SABR model’s ability to fit a volatility smile using different methods of estimation and parametrization. We find that the SABR model is very capable of fitting an observed volatility smile, seemingly regardless of choice of estimation and parametrization method. However, subsequently we note that the risk measure that arises from the SABR model is very much dependent on the parametrization. We analyze this problem and give a correction to the measure, which marginalizes the effect of the choice of parametrization, thus causing the SABR model to yield fairly similar measures for different parametrizations. Finally, we show how the SABR model’s ability to inter- and extrapolate a volatility smile can be utilized in a pricing scenario to price a constant maturity swap. Initially, we explain the theory behind the pricing, and following we present a detailed numerical example using market data and comparing our findings to market prices.

Educations MSc in Mathematics , (Graduate Programme) Final Thesis English 2011 101